Relative equilibria of point vortices. (Aref Memorial Lecture)
Seminar Room 1, Newton Institute
A relative equilibrium of a system of point vortices is a configuration which rotates with constant angular velocity around its centre of vorticity. It is easy to write down the equations for the vortex positions and many simple configurations with symmetry are known. Several asymmetric states have been found numerically, including some surprising ones with some of the vortices being very close. Very little is known analytically about the general problem.
Here we consider the case where the vortices are identical and placed on two perpendicular lines which we choose to be the axes of a coordinate system. We define two polynomials p(z) and q(z) whose roots are the vortex positions on each line in the complex plane, and derive a differential equation for p for given q. We discuss how the general solution to the differential equation relates to physical vortex configurations. The main result is that if q has m solutions symmetrically placed relative to the real axis and p is of degree n, it must have at least n-m+2 real roots. For m=2 this is a complete characterisation, and we obtain an asymptotic result for the location of the two vortices on the imaginary axis as the number of vortices on the real axis tends to infinity.